Showing papers in "Kodai Mathematical Seminar Reports in 1973"

Invariant submanifolds of an f -manifold with complemented frames

Abstract: In an w-dimensional differentiate manifold M of class C°°, a tensor field / of type (1,1), which satisfies /-t-/=0 and is of constant rank r at each point of M, is called an f-structure of rank r and M with an /-structure an f-manifold ([9], [10]). The tensor fields —/ and / + l, 1 being the unit tensor field, are complementary projection operators which define two complementary distributions in M corresponding to the projection operators —f and / + l respectively. The distribution corresponding to —f is r-dimensional and that corresponding to /+l (m—r)-dimensional. The /-manifolds have been studied by Ishihara and the present author [3], [11]. If there exist m — r vector fields Ua ( 2, •••, m — r) spanning the distribution corresponding to /+l and m—r 1-forms u satisfying

Submanifolds umbilical with respect to a non-parallel normal subbundle

Abstract: Let Vn be an n-dimensional submanifold of an ra-dimensional Riemannian manifold Vm and C be a unit normal vector field of Vn in F m If the second fundamental tensor in the normal direction C is proportional to the first fundamental tensor of the submanifold Fw, then Vn is said to be umbilical with respect to the normal direction C Let N be a subbundle of the normal bundle of Vn in VmIf the submanifold Vn is umbilical with respect to every normal direction in N, then Vn is said to be umbilical with respect to N If the covariant derivative of every unit normal direction in N has no component in the complementary normal subbundle N orthogonal to N, then the subbundle N is said to be parallel If there exists, in N, a unit normal direction C such that the covariant derivative of C has nonzero component in the subbundle N everywhere, the subbundle is said to be non-parallel In this paper, we shall study submanifolds of a space form which are umbilical with respect to a non-parallel normal subbundle

On the growth of algebroid functions of finite lower order

Abstract: This conjecture was proved by Valiron [7] for λ